A non-Hermitian Su-Schrieffer-Heeger model with the energy levels of free parafermions

TL;DR

We develop a framework to construct non-Hermitian tight-binding models with $p$ orbitals per unit cell from Hermitian bipartite parents by introducing unidirectional hopping, enforcing a complex chiral symmetry $Z\mathcal{H}(k)Z^{-1}=\omega\mathcal{H}(k)$ with $\omega=e^{2\pi i/p}$. Fully unidirectional hopping yields $p$ complex bands related by $p$-th roots of unity and a common $k$-dependent real factor from the parent, with the SSH parent reproducing the single-particle levels of Baxter’s non-Hermitian clock model parafermions. Partial unidirectionality ($0<u<1$) breaks inversion symmetry and mixes parity blocks, driving a real-to-complex spectrum evolution that depends on the parent topology and the orbital count; edge states and soliton states can remain real under certain conditions, while exceptional-point physics emerges at edges and domain walls. Extending the construction to graphene, higher-order root models of Dirac physics arise, featuring Dirac-point exceptional points and Berry phases, and generalizations address nonuniform couplings, the non-Hermitian skin effect, and small deviations from perfect unidirectionality. Overall, the work links non-Hermitian root constructions to parafermion spectra and topology, with potential realizations in photonic, circuit-based, and cold-atom platforms.

Abstract

Using a parent Hermitian tight-binding model on a bipartite lattice with chiral symmetry, we theoretically generate non-Hermitian models for free fermions with $p$ orbitals per unit cell satisfying a complex generalization of chiral symmetry. The $p$ complex energy bands in $k$ space are given by a common $k$-dependent real factor, determined by the bands of the parent model, multiplied by the $p$th roots of unity. When the parent model is the Su-Schrieffer-Heeger (SSH) model, the single-particle energy levels are the same as those of free parafermion solutions to Baxter's non-Hermitian clock model. This construction relies on fully unidirectional hopping to create Bloch Hamiltonians with the form of generalized permutation matrices, but we also describe the effect of partial unidirectional hopping. For fully bidirectional hopping, the Bloch Hamiltonians are Hermitian and may be separated into even and odd parity blocks with respect to inversion of the orbitals within the unit cell. Partially unidirectional hopping breaks the inversion symmetry and mixes the even and odd blocks, and the real energy spectrum evolves into a complex one as the degree of unidirectionality increases, with details determined by the topology of the parent model and the number of orbitals per unit cell, $p$. We describe this process in detail for $p=3$ and $p=4$ with the SSH model. We also apply our approach to graphene, and show that $AA$-stacked bilayer graphene evolves into a square root Hamiltonian of monolayer graphene with the introduction of unidirectional hopping. We show that higher-order exceptional points occur at edge states and solitons in the non-Hermitian SSH model, and at the Dirac point of non-Hermitian graphene.

Figures (13)

• Figure 1: (a) Hermitian SSH model with two orbitals per unit cell on sublattices $A$ and $B$ with intracell hopping $t \geq 0$ and intercell hopping $J \geq 0$. (b) Non-Hermitian Hamiltonian formed from the parent SSH model with two orbitals $B1$ and $B2$ connected by unidirectional hopping $\gamma > 0$ as indicated by the arrows, with unidirectional hopping from $A$ to $B1$ and from $B2$ to $A$. (c) The topologically trivial phase in the trimer limit with $J = 0$, where each trimer has three states with energies $\epsilon^3 = \gamma t^2$. (d) The topologically nontrivial phase in the trimer limit with $t = 0$, where each trimer has three states with energies $\epsilon^3 = \gamma J^2$, and there are three edge states with energy $\epsilon = 0$.
• Figure 2: Complex energy spectra for $p=3$ orbitals, model $H^{(1,2)}$, as a function of the degree of unidirectionality $u$ when the SSH model is the parent model. The top row shows energy eigenvalues (circles) determined numerically in position space by diagonalizing ${\cal H}^{(1,2)} (u) = {\cal H}^{(1,2)} (1) + (1-u) ( {\cal H}^{(1,2)} (1) )^{\dagger}$ where ${\cal H}^{(1,2)} (1)$ is given in Eq. (\ref{['hp3']}), using open boundary conditions and $L = 200$ unit cells. The inset in (b) shows a close up of the region where the eigenvalues form a circular shape. For all $u$ values, there are three edge states with energies on the real axis (isolated circles, except one is obscured by other energies in (a) and (b)), and they are threefold degenerate at zero energy for $u=1$ in (e). The second row shows the real part of the energy bands and the third row shows their imaginary part, plotted for $-\pi \leq k a \leq \pi$ and obtained by diagonalizing the Bloch Hamiltonian (\ref{['blochh12']}). Dashed lines show the band which is always real, and solid lines show the two bands which are partly real and partly imaginary. When the imaginary parts of the latter (solid lines) are non-zero, their real parts are superimposed on each other and appear as a single line in the plots. The bottom row shows the response power $P(\epsilon)$ (\ref{['powerdef']}) as a function of real energy $\epsilon$ determined in position space using open boundary conditions and $L = 200$ unit cells. To smooth these plots, we add a small imaginary energy as $\epsilon + i \delta$ where $\delta = 0.005$. In all plots, parameter values are $t = 0.5$ and $J = \gamma = 1.0$.
• Figure 3: Non-Hermitian Hamiltonians formed from the parent SSH model with $p=4$ orbitals per unit cell. (a) Model $H^{(1,3)}$ with three orbitals $B1$, $B2$, $B3$ connected by unidirectional hopping $\gamma > 0$ as indicated by the arrows, with unidirectional hopping from $A$ to $B1$ and from $B3$ to $A$. (b) Model $H^{(2,2)}$ with two orbitals $A1$, $A2$ and two orbitals $B1$, $B2$ connected by unidirectional hopping $\gamma > 0$, with unidirectional hopping from $A2$ to $B1$ and from $B2$ to $A1$. (c) Model $H^{(2,2)}$ with a domain wall hosting localized states.
• Figure 4: Complex energy spectra for $p=4$ orbitals, as a function of the degree of unidirectionality $u$ when the SSH model is the parent model. The top row shows model $H^{(1,3)}$, the bottom row $H^{(2,2)}$. Energy eigenvalues (circles) are determined numerically in position space with open boundary conditions by diagonalizing the Hamiltonian. Parameter values are $t = 0.5$, $J = \gamma = 1.0$, and there are $L = 200$ unit cells. For all $u$ values, there are four edge states with energies on the real axis (isolated circles). For $H^{(1,3)}$ (top row), two are degenerate at zero energy, the other two are at non-zero energy for $u < 1$. For $H^{(2,2)}$ (bottom row), the edge states are twofold degenerate for all $u < 1$. For $u=1$, the edge states are fourfold degenerate at zero energy in (e) and (j).
• Figure 5: Complex energy spectra (top row) and the response power $P(\epsilon)$ (bottom row) with a domain wall in $H^{(2,2)}$ when the SSH model is the parent model. All data is determined numerically in position space with open boundary conditions by diagonalizing the Hamiltonian. Parameter values are $t = \gamma = 1.0$, $J = 0.5$, and there are $802$ orbitals. We consider strong $t$ bonds at the edges and a domain wall at the center of the system. For the energy spectra (top row), states localized on the domain wall are isolated circles on the real energy axis, and they are twofold degenerate at zero energy for $u=1$. The bottom row shows the response power $P(\epsilon)$ (\ref{['powerdef']}) as a function of real energy $\epsilon$ where we add a small imaginary energy as $\epsilon + i \delta$ with $\delta = 0.005$.